Mathematics

Problem

Let

f(x)=\ln(1+3x)

, defined for

|x|<\dfrac13

. A Taylor polynomial centered at

x=0

(a Maclaurin polynomial) will be used to approximate

f

for small values of

x

. (a) Calculate

f^{\prime}(0)

f^{\prime\prime}(0)

, and

f^{(3)}(0)

. Show all differentiation steps clearly. (b) Write the third–degree Maclaurin polynomial

P_3(x)

for

f(x)

. (c) Use

P_3(x)

to approximate

f(0.05)

. Report the numerical result to six decimal places.